If \( \frac{\mathrm{1} }{\mathrm{3} } \) of an office staff is on vacation and \( \frac{\mathrm{1} }{\mathrm{4} } \) of those not on vacation are out sick, what fraction of the staff is in the office?

If \( \frac{\mathrm{1} }{\mathrm{3} } \) of an office staff is on vacation and \( \frac{\mathrm{1} }{\mathrm{4} } \) of those not on vacation are out sick, what fraction of the staff is in the office?

Detailed Explanation

If 1/3 are on vacation, then \(\frac{\mathrm{2} }{\mathrm{3} }\) are not. Of those \( \frac{\mathrm{2} }{\mathrm{3} } \) not on vacation, \(\frac{\mathrm{3} }{\mathrm{4} }\) are NOT sick. Therefore \( \frac{\mathrm{2} }{\mathrm{3} } \times \frac{\mathrm{3} }{\mathrm{4} } \) are in the office (not on vacation and not sick) = \((2 \times 3) \div (3 \times 4) = \frac{\mathrm{6} }{\mathrm{12}} = \frac{\mathrm{1} }{\mathrm{2} } \)